Differential Equations MCQ Quiz - Objective Question with Answer for Differential Equations - Download Free PDF

Ans : (4)

Solution :

Let y = 

∵ 

⇒ 

Equating coefficient of x^j to zero; j(j - 1)a_j + (j + 1)a_j+1 - 4a_j = 0

⇒ (j + 1)aj+1 = 4aj - j(j - 1)aj ⇒ aj+1 = 

Calculation: 

So, 

Hence, the correct answer is Option 1. 

Explanation:

 + 2y sec²x = 2sec²x + 3 tan x sec²x 

I.F. = 

I.F. = e^2tanx

Put tan x = u

sec²xdx = du 

F(0) = 

1 = C

 = 21

Calculation: 

⇒ 

Let 

∴ 

⇒ 

y(1) = 3

Hence, the correct answer is Option 1. 

Concept:

Transformation of Variable in Second Order Differential Equations:

• A second-order linear differential equation with variable coefficients can sometimes be transformed into one with constant coefficients by a suitable change of variables.
• Given equation: d²y/dx² + P(x) dy/dx + e^2x y = 0
• Let the new variable be s = s(x) such that the new equation in s has constant coefficients.
• Using chain rule for derivatives:
  • dy/dx = dy/ds × ds/dx
  • d²y/dx² = d²y/ds² × (ds/dx)² + dy/ds × d²s/dx²
• Substituting into the equation yields transformed coefficients in terms of ds/dx and d²s/dx².
• To achieve constant coefficients, expressions involving P(x) and e^2x must cancel appropriately.

Calculation:

Given,

d²y/dx² + P(x) dy/dx + e^2x y = 0

Let s = s(x), such that ds/dx = e^x

⇒ dy/dx = dy/ds × e^x

⇒ d²y/dx² = d²y/ds² × e^2x + dy/ds × e^x

Substitute in original equation:

⇒ e^2x d²y/ds² + e^x dy/ds + P(x) e^x dy/ds + e^2x y = 0

⇒ e^2x d²y/ds² + e^x (1 + P(x)) dy/ds + e^2x y = 0

To make coefficients constant:

⇒ e^x(P(x) + 1) must be constant

⇒ e^−x(P(x) + 1) = constant

∴ e^−x(P(x) + 1) is a constant function on ℝ.

Concept:

Order: The order of a differential equation is the order of the highest derivative appearing in it.

Degree: The degree of a differential equation is the power of the highest derivative occurring in it, after the equation has been expressed in a form free from radicals as far as the derivatives are concerned.

Calculation:

Given:

For the given differential equation the highest order derivative is 1.

Now, the power of the highest order derivative is 3.

We know that the degree of a differential equation is the power of the highest derivative

Hence, the degree of the differential equation is 3.

Mistake PointsNote that, there is a term (dx/dy) which needs to convert into the dy/dx form before calculating the degree or order. 

Concept:

Order: The order of a differential equation is the order of the highest derivative appearing in it.

Degree: The degree of a differential equation is the power of the highest derivative occurring in it, after the Equation has been expressed in a form free from radicals as far as the derivatives are concerned.

Calculation:

The differential equation is given as: 

The highest order derivative presents in the differential equation is 

Hence, its order is three.

Here the given differential equation is not a polynomial equation, Hence its degree is not defined.

Concept:

Calculation:

Given: dy = (1 + y²) dx

Integrating both sides, we get

⇒ y = tan (x + c)

∴ The solution of the given differential equation is y = tan (x + c).

Given:

x = 1

x2 + y2 + z2 = xy + yz + zx

Calculations:

x² + y2 + z2 - xy - yz - zx = 0

⇒(1/2)[(x - y)² + (y - z)² + (z - x)²] = 0

⇒x = y , y = z and z = x

But x = y = z = 1

so, 

= {10(1)⁴ + 5(1)⁴ + 7(1)⁴}/{13(1)²(1)²+ 6(1)²(1)² + 3(1)²(1)²}

= 22/22

= 1

Hence, the required value is 1.

Calculation:

Given: 

On integrating both sides, we get

⇒ y = xe^a + c

Concept:

Order: The order of a differential equation is the order of the highest derivative appearing in it.

Degree: The degree of a differential equation is the power of the highest derivative occurring in it, after the Equation has been expressed in a form free from radicals as far as the derivatives are concerned.

Calculation:

Given:

For the given differential equation the highest order derivative is 1.

Now, the power of the highest order derivative is 3.

We know that the degree of a differential equation is the power of the highest derivative.

Hence, the degree of the differential equation is 3.

Concept:

Calculation:

Given: 

Integrating both sides, we get

Given:

x +  = 3

Concept Used:

Simple calculations is used

Calculations:

⇒ x +  = 3

On multiplying 2 on both sides, we get

⇒ 2x +  = 6  .................(1)

Now, On cubing both sides,

⇒ 

⇒ 

⇒ 

⇒ 

⇒   ..............from (1)

⇒ 

⇒ 

⇒ Hence, The value of the above equation is 180

Concept:

Order: The order of a differential equation is the order of the highest derivative appearing in it.

Degree: The degree of a differential equation is the power of the highest derivative occurring in it, after the Equation has been expressed in a form free from radicals as far as the derivatives are concerned.

Calculation:

For the given differential equation the highest order derivative is 2.

The given differential equation is not a polynomial equation because it involved a logarithmic term in its derivatives hence its degree is not defined.

Concept: 

Calculation: 

Given :  

⇒  

Integrating both sides, we get 

⇒  

⇒ 

⇒   [∵ 2c = C]

⇒ 

The correct option is 2 .